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Carnot theory
"Carnot's theorem, developed in 1824 by Nicolas Léonard Sadi Carnot, also called Carnot's rule, is a principle that specifies limits on the maximum efficiency any heat engine can obtain. The efficiency of a Carnot engine depends solely on the difference between the hot and cold temperature reservoirs." Carnot theory is an extension and generalisation of Carnot's work on thermodynamic efficiency. By extending thermodynamics theory further into information technology we develop language to describe program-efficiency in computer science. Carnot's Theorem The formula for this maximum efficiency is : \eta_{\text{max}} = \eta_{\text{Carnot}} = 1 - \frac{T_\mathrm{C}}{T_\mathrm{H}} Where T means TEMPERATURE for thermodynamics The C and H represent COLDEST and HOTTEST temperatures in thermal contact within an engine The fraction T_C/T_H then asks: * How many times less hot than T_H is T_C? e.g. if T_C=100 K and T_H=200 K, then T_C/T_H=1/2 - T_C is half as hot as T_H In the above example, the "Carnot efficiency" for such an engine is 50%. Invariability Notice that only the ratio T_C/T_H mattered, not the particular values. The temperatures could have been 1 K and 2 K and the same result would have emerged (or 30 K and 60 K, etc) Limits A fraction A/B for real numbers A & B can be anything from negative infinity to infinity. However, with T_C/T_H there is an implicit meaning that T_H>=T_C (the "hot" temperature is not colder than the "cold" one). Hence, the ratio is bounded by 1, since T_C can never surpass equality with T_H. Since efficiency H=1-(T_C/T_H) then H=1-1 = 0 So, an engine operating between two equal temperatures has a maximum efficiency of zero: it cannot produce useful energy. The best case occurs when H=1-(T_C/T_H)= 1 - 0 = 1 Hence, the ideal scenario is for T_C/T_H = 0 (T_C = 0 and/or T_H -> infinity) - note: we need T=0 K to be the ABSOLUTE ZERO temperature for this to work, hence we use Kelvins not Celsius. Cryptographic Interpretation In the context of information flow instead of energy flow, Work shifts from meaning "useful energy" to meaning "useful information". Temperature maintains it's meaning in the sense of a measure of the "thermal noise" of a medium. Thermodynamic engines try to convert a high-energy source into a useful output by selective coupling to a low-noise "sink". A decryption algorithm attempts to convert a high-noise source of information into a high-information output through translations of the data that follow a low-noise algorithm. Quantum Cryptography Unlike classical cryptography, communication errors in Quantum Computing cannot be deterministically corrected, and hence quantum algorithms are generally limited to giving solutions probabilistically. Of course, even for classical computers there are finite probabilities of a colossal bit error than exceeds their error processing capability - but the probability of enough errors occurring across multiple bits are so astronomical that we treat them as deterministic computation devices. Carnot's theory allows us to interrogate the true underlying process of data conversion: Quarius' Theorem of Universal Decryption H = 1 - σ_out/σ_in The maximum efficiency of a decryption protocol is equal to 1 minus the ratio of the output uncertainty (σ_out) to the input uncertainty (σ_in). Examples If a decryption protocol is deterministic and lossless then the output uncertainty of the decrypted data is 0: σ_out = 0 The only remaining variable is σ_in - the uncertainty in the data's meaning prior to decryption. If σ_out = σ_in - Δσ - where Δσ is the uncertainty reduction during the decryption Then we have: H = 1 - (σ_in - Δσ)/(σ_in) = 1 - (1-Δσ/σ_in) = Δσ/σ_in So, the efficiency of a decryption approaches 0% as the uncertainty reduction approaches zero, and approaches 100% as the reduction approaches the input uncertainty. Notes for a future Seph: * Non-deterministic decryption theory * Given a quantum process has finite error rate, the most efficient decryption method is not a lossless one. Some loss will occur regardless, so the most efficient decryption is that which removes error rates the most for rates above that noise-floor of the processing system. * So if the bit error rate for an RBG image of 360,000 pixels (600x600) when vieing on a laptop screen is 10% change per pixel and an undetectable "random pixel" when the total randomized noise affects less than 1 in 10,000 pixels (~36 pixels wrong out of the 360k image). * If we could store enough information in that single image such that only a byte could be used per pixel? yes - if you compile it in a way that knows how to order the data into pixels and interpret the colour and brightness qualities of the pixel from that single byte of data (2^8=256 colour image). * Then, if we do so, and subsequently add a second input byte per pixel to the code - that second byte is free to carry non-visual information. * In analogy to DNA, the image itself may be generated through the code of the second byte. And through analysing many millions of images generated by the same secondary-byte, a cryptographer may be able to observe patterns in the produced images to prove that the secondary byte must follow a particular style of alogrithm and hence predict with some finite probability what the secondary byte encodes. * For example a pixel might have 8 different colour options (white, black, red, green, blue, yellow, orange, purple) and have a value from 0-7 for that pixel's first byte. * The second byte then may encode how to order the pixels. For a rectangular image, the bit 0 might indicate to place the next pixel to the right of this pixel, the bit 1 might indicate to place it in the first column of pixels, forming a new row. or if two bits are used then those could be 00 and 01, while 10 could indicate you place it to a new column at the end of the row, and 11 could indicate placing the next pixel to the left. * 2 bits is only quarter of a byte, when you take 2 bits and add another 2, you get 4 bits - another 4 and you got a byte. complexity is exponential. So the other 2 bits can encode an extra piece of information such as translations in phase for a holographic image. 00-encoded pixels might have a no Ghost image, 01- encoded pixels might have a lagging ghost, 10-encoded pixels might have a leading ghost and 11-encoded pixels might remain persistent through the animation/hologram. * Instead by going to one byte-each, the colour can be expressed from a set of 256 colours and the second bit can express the phase co-ordinate of this pixel in the first frame of some 256-frame animation of the image. Hence you will have encoded a complex animation, gaining a 25,000% increase of frame number while only a 100% increase in the file-size. :) * The compiler used then applies it's algorithm to adjust the appearance of the image in accordance with some Hamiltonian that propagates the phase of each pixel and adjusts the colour value accordingly. Category:Thermodynamics Category:Computers Category:Information Theory Category:Engineering